Smith set

In voting systems, the Smith set, named after John H. Smith, but also known as the top cycle, or as GETCHA, is the smallest non-empty set of candidates in a particular election such that each member defeats every other candidate outside the set in a pairwise election. The Smith set provides one standard of optimal choice for an election outcome. Voting systems that always elect a candidate from the Smith set pass the Smith criterion and are said to be "Smith-efficient".

A set of candidates where every member of the set pairwise defeats every member outside of the set is known as a dominating set.

The Schwartz set is closely related to and is always a subset of the Smith set. The Smith set is larger if and only if a candidate in the Schwartz set has a pair-wise tie with a candidate that is not in the Schwartz set.

The Smith set can be constructed from the Schwartz set by repeatedly adding two types of candidates until no more such candidates exist outside the set:

Note that candidates of the second type can only exist after candidates of the first type have been added.

Any binary relation ${\displaystyle R}$ on a set ${\displaystyle A}$ can generate a natural partial order on the ${\displaystyle R}$-cycle equivalence classes of set ${\displaystyle A}$, so that ${\displaystyle xRy}$ implies ${\displaystyle [x]\geq [y]}$.

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